We solve the examples considered using the preconditioned GMRES or flexible GMRES (FGMRES) [1]} methods in conjunction with the preconditioners of Method I, Method II, Method III and the SNSS method.
In Examples  and  the main system is solved using the FGMRES method and in Example  by the GMRES method.
The outer iteration (GMRES of FGMRES) is stopped as soon as the Euclidean norm of the residual of original system is reduced by a factor of \(10^{10}\)  and the maximum number of iterations is set to be 1000.
We always solve the inner systems using the Chebyshev semi-iteration method with the PRESB preconditioner.
In Examples  and  the iteration of Chebyshev semi-iteration method is terminated as soon as the Euclidean norm of the residual of the un-preconditioned system is reduced by a factor of \(10^2\)  and in Example  by a factor of \(10^{10}\) . The maximum number of iteration of Chebyshev semi-iteration is set to be 20. We always use a null vector as an initial guess. It is noted that the innermost subsystems are solved exactly using the sparse Cholesky factorization incorporated with the symmetric approximate minimum degree permutation. To do so, the symamd.m command of Matlab is applied.
